Hypersurfaces of a sphere with parallel shape operator

Abstract

In [53] P. J. Ryan considered hypersurfaces of real space forms and specifically, he gave a complete classification of hypersurfaces in the sphere which satisfy a certain condition. The condition that the shape operator is parallel is its special case. In this section we give the proof of this classification (in the specific case $\nabla_XA=0$) and furthermore, we show that the algebraic condition (13.5) on the shape operator implies that it is parallel.